Convolutional and computer vision methods for accelerating partial tracing operation in quantum mechanics for general qudit systems

Partial trace is a mathematical operation used extensively in quantum mechanics to study the subsystems of a composite quantum system and in several other applications such as calculation of entanglement measures. Calculating partial trace proves to be a computational challenge with an increase in the number of qubits as the Hilbert space dimension scales up exponentially and more so as we go from two-level systems (qubits) to D-level systems. In this paper, we present a novel approach to the partial trace operation that provides a geometrical insight into the structures and features of the partial trace operation. We utilize these facts to propose a new method to calculate partial trace using signal processing concepts, namely convolution, filters and multigrids. Our proposed method of partial tracing significantly reduces the computational complexity by directly selecting the features of the reduced subsystem rather than eliminating the traced-out subsystems. We give a detailed description of our method and provide some explicit examples of the computation. Our method can be generalized further to a D-level system of N-particles with a considerable reduction in computation time. The arithmetic complexity of our algorithm is \(\mathcal {O}\left( D^{2N - n}\right) \) with n subsystems partially traced out. We also observe various geometrical patterns and self-forming fractal structures, which we discuss here. We give numerical evidence to all the claims.